Assuming $x-y$ axes on the normal scale we can count primes using $\pi(x).$ We know that a complex function's zeros encode information about the distribution of $\pi(x),$ and that is the Riemann zeta function, $\zeta(s).$
Does counting primes on a different scale imply that there's a complex function whose zeros encode the distribution of those primes? If so, how would you derive the complex function? Are they all just some transformation of $\zeta(s)$?
Below I've outlined two ways to count primes on two different scales:
Say you wanted to count numbers $e^p$ for $p\in \Bbb P$ less than a certain amount. You decide to count them in the following way: $f(e^2)=e^1,$ $f(e^3)=e^2,$ and $f(e^4)=e^2, f(e^5)=e^3.$
Alternatively you want to count numbers $\ln(p)$ for $p\in \Bbb P$ less than a certain amount. And you count them in the following way: $g(\ln(2))=\ln(1), g(\ln(3))=\ln(2), g(\ln(4))=\ln(2)$ and $g(\ln(5))=\ln(3).$